Spread 1 cup sauce in an 8-inch square baking dish. Top with 2 noodles. Layer with half the artichokes, then 1 cup sauce and 1/4 cup Parmesan. Top with 2 more noodles, half the mushrooms, 1 cup sauce, and 1/4 cup Parmesan. Repeat layering with remaining ingredients, reserving the last 1/4 cup Parmesan. Cover lasagna loosely with foil and bake on top rack until sauce is bubbling, 35 to 40 minutes. (Place a rimmed baking sheet on middle rack to catch any drippings.) Remove lasagna from oven and heat broiler. Sprinkle with reserved Parmesan and broil until cheese is browned and bubbling, 3 to 5 minutes. Let cool 20 minutes before slicing.